Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions

Abstract

In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of φ 4 models with either nearest-neighbours and mean-field interactions.

The geometric theory of phase transitions

“…It is worth noting that our results are fully in agreement with those obtained in Ref. [28]. Surprisingly, Eqs.…”

“…Nevertheless, such an idea has been recently developed by Franzosi et al in Ref. [28]: their results show that the differential geometry is undoubtedly a powerful and reliable tool for investigating PTs in the microcanonical ensemble. In a last paper [31], instead, we have shown that, adopting a revised definition of entropy as suggested by Franzosi [29,30], one can identify each derivative of the entropy with a specific geometric structure.…”

“…To this purpose, we adopt the Ehrenfest-like classification developed in Ref. [28] for the microcanonical ensemble and we make explicit the dependence of any observable by the number of DoF, n. Essentially, it is well-known that the canonical free energy, f n (β), with β = 1/k B T , is related to the microcanonical entropy through the Legendre transform:…”

Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions

“…To the contrary, the set of uncoupled harmonic oscillators is integrable, however, each single harmonic oscillator is ergodic in its own two-dimensional phase space, and, since all the oscillators have the same frequency, so that they are interchangeable, and the initial conditions are random, also this systems behaves as if it was ergodic, as the stability of the results of the computation of the geometric observables has been checked by changing the initial conditions. In fact, given an observable, Φ, defined on the phase space, the microcanonical averages can be measured along the dynamics as follows [23,34]:…”

“…It is worth mentioning that in reference [34], similar investigations were proposed concerning the relation between geometric and thermodynamic quantities and their behavior in the presence of a PT; in this context, the derivatives of the entropy are treated as observables, such as temperature, specific heat and the second order derivative of the entropy. Contrarily, our present approach treats S as an unknown function which can be determined by solving Equation ( 40).…”